Subgroups — #28373160

Order 13952.

380 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295580	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20295358	C₂	1	1	yes	maximal, characteristic
4	#20295355	C₂	1	1	yes	maximal, characteristic
5	#20295310	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295309	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295235	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295234	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295183	C₂	1	1	yes	maximal, characteristic
10	#20295180	C₂	1	1	yes	maximal, characteristic
11	#20295134	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295131	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295131	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295081	C₂	1	1	yes	maximal, characteristic
15	#441×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
18	C₂×#13182019	C₂²	1	1	yes	characteristic
19	C₂×#13182019	C₂²	1	1	yes	characteristic
20	#16812698	C₂²	1	1	yes	characteristic
21	#16812697	C₂²	1	1	yes	characteristic
22	#16812689	—	2	1	no	—
23	#16812679	C₂²	1	1	yes	characteristic
24	#16812677	—	2	1	no	—
25	#16812675	C₂²	1	1	yes	characteristic
26	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
27	C₄×Dic₂₁₈	—	2	1	no	—
28	C₄×Dic₂₁₈	—	2	1	no	—
29	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
31	#16812664	—	2	1	no	—
32	#16812664	C₂²	1	1	yes	characteristic
33	#16812656	—	2	1	no	—
34	C₄⋊C₄×D₁₀₉	—	2	1	no	—
35	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
36	C₄²×D₁₀₉	—	2	1	no	—
37	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
38	#16812631	C₂²	1	1	yes	characteristic
39	#16812630	C₂²	1	1	yes	characteristic
40	#16812630	C₂²	1	1	yes	characteristic
41	#16812628	C₂²	1	1	yes	characteristic
42	#16812628	C₂²	1	1	yes	characteristic
43	#16812624	C₂²	1	1	yes	characteristic
44	#16812624	C₂²	1	1	yes	characteristic
45	#16812623	C₂²	1	1	yes	characteristic
46	#16812623	C₂²	1	1	yes	characteristic
47	#16812622	C₂²	1	1	yes	characteristic
48	#16812622	C₂²	1	1	yes	characteristic
49	#16812611	C₂²	1	1	yes	characteristic
50	#16812611	C₂²	1	1	yes	characteristic
51	#16812610	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	#97×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×Dic₂₁₈	—	2	1	no	—
61	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
62	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
63	Q₈×D₁₀₉	—	2	2	no	—
64	Q₈×D₁₀₉	—	2	2	no	—
65	C₂×C₄×D₁₀₉	—	2	1	no	—
66	C₂×C₄×D₁₀₉	—	2	1	no	—
67	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
70	Dic₄₃₆	—	2	2	no	—
71	#13182019	—	2	2	no	—
72	#13182019	—	2	2	no	—
73	C₂×#11724784	C₂³	1	1	yes	characteristic
74	C₂×#11724784	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₀₉	—	2	1	no	—
76	C₄⋊Dic₁₀₉	—	2	1	no	—
77	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	#13182008	—	2	1	no	—
86	#13182008	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	C₄×Dic₁₀₉	—	2	1	no	—
90	C₄×Dic₁₀₉	—	2	1	no	—
91	C₄×Dic₁₀₉	—	2	1	no	—
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
94	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
95	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	Q₁₆×C₁₀₉	—	2	2	no	—
99	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₀₉	—	2	1	no	—
102	C₄⋊C₄×C₁₀₉	—	2	1	no	—
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄²×C₁₀₉	—	2	1	no	—
106	C₄²×C₁₀₉	—	2	1	no	—
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₁₀₉	—	2	1	no	—
110	Q₈×C₁₀₉	—	2	1	no	—
111	Q₈×C₁₀₉	—	2	1	no	—
112	Q₈×C₁₀₉	#41	1	2	yes	—
113	C₂×C₄×C₁₀₉	—	2	1	no	—
114	C₂×C₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
120	C₂×Dic₁₀₉	—	2	1	no	—
121	C₂×Dic₁₀₉	—	2	1	no	—
122	C₂×Dic₁₀₉	—	2	1	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₄×D₁₀₉	—	2	4	no	—
128	C₄×D₁₀₉	—	2	2	no	—
129	C₄×D₁₀₉	—	2	1	no	—
130	C₄×D₁₀₉	—	2	1	no	—
131	C₄×D₁₀₉	—	2	2	no	—
132	C₄×D₁₀₉	—	4	2	no	—
133	Dic₂₁₈	—	2	4	no	—
134	Dic₂₁₈	—	2	1	no	—
135	Dic₂₁₈	—	2	1	no	—
136	Dic₂₁₈	—	2	1	no	—
137	Dic₂₁₈	—	4	2	no	—
138	Dic₂₁₈	#41	1	2	yes	—
139	C₈×C₁₀₉	—	2	1	no	—
140	C₈×C₁₀₉	—	2	1	no	—
141	#11724784	—	2	1	no	—
142	#11724784	—	2	1	no	—
143	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₁₀₉	—	2	1	no	—
145	C₂×D₁₀₉	—	2	1	no	—
146	C₂×D₁₀₉	#137	1	2	yes	—
147	C₄×C₁₀₉	—	2	2	no	—
148	C₄×C₁₀₉	—	2	1	no	—
149	C₄×C₁₀₉	—	2	1	no	—
150	C₄×C₁₀₉	—	4	1	no	—
151	C₄×C₁₀₉	—	4	1	no	—
152	C₄×C₁₀₉	—	4	1	no	—
153	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
154	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
155	Dic₁₀₉	—	2	1	no	—
156	Dic₁₀₉	—	2	2	no	—
157	Dic₁₀₉	—	2	1	no	—
158	Dic₁₀₉	—	4	1	no	—
159	Dic₁₀₉	—	4	1	no	—
160	Dic₁₀₉	—	4	1	no	—
161	Dic₁₀₉	#137	1	2	yes	—
162	C₂×C₁₀₉	#525	1	1	yes	characteristic
163	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
164	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
165	D₁₀₉	—	2	2	no	—
166	#3113	—	109	1	no	maximal, Sylow
167	C₁₀₉	#3113	1	1	yes	characteristic, complemented, Sylow
168	#533	—	109	1	no	—
169	C₂×C₄×Q₈	—	109	1	no	—
170	#483	—	109	1	no	—
171	#483	—	109	1	no	—
172	#475	—	109	1	no	—
173	#475	—	109	1	no	—
174	#462	—	109	1	no	—
175	#462	—	109	1	no	—
176	#451	—	109	1	no	—
177	#451	—	109	1	no	—
178	#441	—	109	1	no	—
179	#441	—	109	1	no	—
180	#441	—	109	1	no	—
181	#441	D₁₀₉	1	1	yes	characteristic, complemented
182	#432	—	109	1	no	—
183	C₂²×Q₈	—	109	1	no	—
184	C₂×Q₁₆	—	109	1	no	—
185	C₂×Q₁₆	—	109	1	no	—
186	C₂×Q₁₆	—	109	1	no	—
187	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
188	C₄⋊Q₈	—	109	1	no	—
189	C₄⋊Q₈	—	109	1	no	—
190	#125	—	218	1	no	—
191	C₂²⋊Q₈	—	109	1	no	—
192	C₂²⋊Q₈	—	109	1	no	—
193	C₂²⋊Q₈	—	218	1	no	—
194	C₄×Q₈	—	109	1	no	—
195	C₄×Q₈	—	109	1	no	—
196	C₄×Q₈	—	109	1	no	—
197	C₄×Q₈	—	109	1	no	—
198	C₄×Q₈	—	218	1	no	—
199	C₄×Q₈	—	218	1	no	—
200	C₄×Q₈	—	218	1	no	—
201	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
202	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
203	#117	—	218	1	no	—
204	C₂×C₄⋊C₄	—	109	1	no	—
205	C₂×C₄⋊C₄	—	218	1	no	—
206	C₂×C₄²	—	109	1	no	—
207	C₂×C₄²	—	218	1	no	—
208	#106	—	109	1	no	—
209	#106	—	109	1	no	—
210	#106	—	109	1	no	—
211	#106	C₂×D₁₀₉	1	1	yes	characteristic
212	C₄⋊C₈	—	109	1	no	—
213	C₄⋊C₈	—	109	1	no	—
214	#103	—	109	1	no	—
215	#103	—	109	1	no	—
216	#103	—	109	1	no	—
217	#103	—	109	1	no	—
218	#103	—	109	1	no	—
219	#103	—	109	1	no	—
220	#103	C₂×D₁₀₉	1	1	yes	characteristic
221	#103	C₂×D₁₀₉	1	1	yes	characteristic
222	C₂²⋊C₈	—	109	1	no	—
223	C₂²⋊C₈	—	109	1	no	—
224	#97	—	109	1	no	—
225	#97	C₂×D₁₀₉	1	1	yes	characteristic
226	C₂×Q₈	—	109	1	no	—
227	C₂×Q₈	—	109	1	no	—
228	C₂×Q₈	—	218	1	no	—
229	C₂×Q₈	—	218	1	no	—
230	C₂×Q₈	—	218	2	no	—
231	C₂×Q₈	—	218	2	no	—
232	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
233	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
234	C₂²×C₄	—	109	1	no	—
235	C₂²×C₄	—	109	1	no	—
236	C₂²×C₄	—	109	1	no	—
237	C₂²×C₄	—	218	1	no	—
238	C₂²×C₄	—	218	1	no	—
239	Q₁₆	—	2	2	no	—
240	Q₁₆	—	218	2	no	—
241	Q₁₆	—	218	2	no	—
242	Q₁₆	—	218	2	no	—
243	C₂×C₈	—	109	1	no	—
244	C₂×C₈	—	109	1	no	—
245	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	109	1	no	—
250	C₄⋊C₄	—	109	1	no	—
251	C₄⋊C₄	—	218	1	no	—
252	C₄⋊C₄	—	218	1	no	—
253	C₄⋊C₄	—	218	1	no	—
254	C₄⋊C₄	—	218	1	no	—
255	C₄⋊C₄	—	218	1	no	—
256	C₄⋊C₄	—	218	1	no	—
257	C₄⋊C₄	—	218	1	no	—
258	C₄⋊C₄	—	218	1	no	—
259	C₄⋊C₄	—	218	1	no	—
260	C₄⋊C₄	—	218	1	no	—
261	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
263	C₂²⋊C₄	—	218	1	no	—
264	C₂²⋊C₄	—	218	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	109	1	no	—
268	C₄²	—	109	1	no	—
269	C₄²	—	109	1	no	—
270	C₄²	—	218	1	no	—
271	C₄²	—	218	1	no	—
272	C₄²	—	218	1	no	—
273	C₄²	—	218	1	no	—
274	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
275	C₂³	—	109	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	109	2	no	—
280	Q₈	—	436	2	no	—
281	Q₈	—	218	4	no	—
282	Q₈	—	218	1	no	—
283	Q₈	—	218	1	no	—
284	Q₈	—	218	1	no	—
285	Q₈	#13182029	1	2	yes	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	109	1	no	—
291	C₂×C₄	—	109	1	no	—
292	C₂×C₄	—	109	1	no	—
293	C₂×C₄	—	436	2	no	—
294	C₂×C₄	—	218	2	no	—
295	C₂×C₄	—	218	2	no	—
296	C₂×C₄	—	218	4	no	—
297	C₂×C₄	—	218	1	no	—
298	C₂×C₄	—	218	1	no	—
299	C₂×C₄	—	218	1	no	—
300	C₂×C₄	—	218	1	no	—
301	C₂×C₄	—	218	1	no	—
302	C₂×C₄	—	218	1	no	—
303	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
304	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	218	1	no	—
309	C₈	—	218	1	no	—
310	C₂²	—	109	2	no	—
311	C₂²	—	218	1	no	—
312	C₂²	—	218	1	no	—
313	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	109	2	no	—
321	C₄	—	436	1	no	—
322	C₄	—	436	1	no	—
323	C₄	—	436	1	no	—
324	C₄	—	218	1	no	—
325	C₄	—	218	2	no	—
326	C₄	—	218	1	no	—
327	C₄	#41×D₁₀₉	1	1	yes	characteristic
328	C₄	C₂×#13182029	1	1	yes	characteristic
329	C₂	—	218	2	no	—
330	C₂	#20295535	1	1	yes	characteristic
331	C₂	#137×D₁₀₉	1	1	yes	characteristic
332	C₂	#137×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#441×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂²×D₁₀₉	C₂×D₄
hypercenter	#441	D₁₀₉
hypocenter	C₁₀₉	#3113
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
5. #20295310	C₂	semidirect
6. #20295309	C₂	semidirect
7. #20295235	C₂	semidirect
8. #20295234	C₂	semidirect
11. #20295134	C₂	semidirect
12. #20295131	C₂	semidirect
13. #20295131	C₂	semidirect
15. #441×C₁₀₉	C₂	semidirect
167. C₁₀₉	#3113	semidirect
181. #441	D₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20295580	2
C₄×Q₈×D₁₀₉	2
#20295358	2
#20295355	2
#20295310	2
#20295309	2
#20295235	2
#20295234	2
#20295183	2
#20295180	2
#20295134	2
#20295131	2
#20295081	2
#441×C₁₀₉	2
#3113	109